The cardinality of sumsets: different summands
Volume 167 / 2015
Acta Arithmetica 167 (2015), 375-395
MSC: Primary 11P99; Secondary 11B30.
DOI: 10.4064/aa167-4-4
Abstract
We offer a complete answer to the following question on the growth of sumsets in commutative groups. Let $h$ be a positive integer and $A, B_1, \dots , B_h$ be finite sets in a commutative group. We bound $|A+B_1+\dots +B_h|$ from above in terms of $|A|$, $|A+B_1|, \dots ,|A+B_h|$ and $h$. Extremal examples, which demonstrate that the bound is asymptotically sharp in all parameters, are furthermore provided.
